A common problem encountered in the operation of electrical machines is cogging. Cogging is the presence of non-linear magnetic torque during the operation of the machine due to the effect of the geometry of the rotor and the stator at the air gap on the flux distribution and the forces between the rotor and the stator. Specifically, it is caused by the rotor having preferred low potential energy positions relative to the stator, where the attraction between the rotor and stator is at a maximum, and corresponding high potential energy positions disposed between each of the low energy positions. It is the difference in potential energy between these positions that determines the magnitude of the cogging torque.
Cogging during the operation of electrical machines can result in decreased efficiency and reliability, as well as causing unwanted vibration and noise and, in extreme cases, complete failure. Cogging is well known in machines in which the stator, the rotor, or both the rotor and the stator exhibit some form of saliency. The effect is also well known in induction machines in which the magnet poles and winding slots interact to cause serious cogging for particular numbers of winding slots. Cogging is particularly pronounced in machines which have a number of winding slots that is an integral multiple of the number of magnet poles and where both the winding slots and magnet poles are uniformly spaced around the circumference of the bodies in which they are formed. This is because, due to the symmetry of such machines, when one magnet pole is in its lowest potential energy position relative to the winding slots, all the other magnet poles will also be in their lowest potential energy position and the rotor will therefore be in the lowest possible potential energy position relative to the stator. Furthermore, this means if there are n winding slots the rotor will have n low potential energy positions relative to the stator and the cogging frequency during operation of the machine will be a product of the rotation frequency of the rotor and n.
Typical permanent-magnet based electrical machines are formed such that the rotor is rotatably mounted within the stator, the magnet poles are uniformly spaced around the circumference of the outer surface of the rotor and the winding slots are uniformly spaced around the circumference of the inner surface of the stator. However, other constructions are also possible. For example, it is possible that the rotor is rotatably mounted outside the stator. It is also possible for the magnet poles to be fixed to the stator and the winding slots to be formed in the rotor. With regards to cogging, the design considerations for permanent-magnet based electrical machines according to any of these constructions are substantially the same.
The factors that must be considered when deciding upon the number of winding slots in an electrical machine are generally well known. For example, it is known that when an electrical machine has a number of winding slots which is an integral multiple of the number of its magnet poles the magneto-motive force (m.m.f.) created by the machine's stator winding during operation will contain a minimized number of harmonics but the machine may also experience pronounced cogging during operation.
Numerous methods of minimizing cogging have been proposed and implemented with varying degrees of success. For some types of machine it is possible to minimize cogging by having a number of winding slots that is not an integral multiple of the number of magnet poles. Such windings are generally referred to as comprising ‘fractional slots per pole per phase’ and are well known to those who are skilled in the art. However, in some cases this may not be preferred, or even possible. For example, in large-scale electrical machines the options for having differing numbers of winding slots that are not multiples of the number of magnet poles are very limited as the non-synchronous flux patterns that result from such changes in the number of winding slots inevitably lead to additional losses in the magnet poles. These losses may be extremely high and even render the method unusable.
An alternative method of reducing cogging is to incorporate magnetic wedges in the winding slots but this method is costly. It is also possible to use ‘semi-closed’ winding slots to reduce cogging but for the types of winding that comprise formed coils and are common in large machines this type of winding slot can lead to an unacceptably high winding cost. Permanent-magnet machines that have magnet poles specifically shaped to reduce cogging have also been proposed. Typically, it is proposed that the outer surfaces of the magnet poles are formed such that the air gap between the stator and the magnet poles is not uniform.
Another method of minimizing cogging torque in permanent-magnet based electrical machines is to have the magnets skewed from their conventional arrangement. For example, Japanese Patent No. 2005-261188 discloses a machine with reduced cogging in which the magnets are skewed such that they are not parallel with the axis of the machine.
A further method of minimizing cogging torque is disclosed in U.S. Pat. No. 4,713,569, which discloses an AC electric motor with a permanent-magnet rotor containing a plurality of magnet poles that are angularly displaced from their reference positions by an amount dependent upon the number of magnet poles and the number of stator poles (winding slots) in the machine. The reference positions of the magnet poles are the positions where they would be situated were they equally spaced circumferentially. Specifically, in electrical machines according to this U.S. patent, each magnet pole is displaced from its reference position by a different amount, the differing amounts all being integral multiples of 360° divided by the product of the number of stator poles (winding slots) and the number of magnet poles, such that no magnet pole is displaced from its reference position by more than the pitch of the stator poles (winding slots). The specific example given in the specification of this U.S. patent is of an AC electric motor with a permanent-magnet rotor with 8 magnet poles and a stator with 24 poles (winding slots) and wherein the magnet poles are displaced from their reference positions by integral multiples of 1.875° (360°/(24×8)) such that no two poles are displaced from their reference positions by the same amount and the (nominal) 8th magnet pole is displaced from its reference position by an amount equal to the stator pole (winding slot) pitch.
The rotor construction disclosed in U.S. Pat. No. 4,713,569 provides a substantial reduction in cogging torque as compared to conventional electrical machines with uniformly spaced magnet poles. However, in most cases this rotor construction will not be preferred to the conventional construction due to its impact on other design considerations. For example, this construction results in complete asymmetry of the positioning of the magnet poles around the rotor. If the electrical machine is intended for high-speed use, then this asymmetry is undesirable as it means the magnet pole arrangement is not mechanically balanced. This complete asymmetry also results in a pronounced harmonic content of either the m.m.f. created by the stator winding if the machine is a motor or the electro-motive force (e.m.f.) waveform created in the stator winding if the machine is a generator.
The rotor construction of U.S. Pat. No. 4,713,569 may also not be preferred due to the separation of the first and last magnet poles as defined in that patent. This is because the angular separation between these two magnet poles is significantly less than in a conventional electrical machine with uniformly spaced magnet poles. Specifically, the separation between these two poles is less than the uniform spacing by an amount equal to the winding slot pitch. Depending upon the angular width of the magnets that form these magnet poles, this could lead to the two magnet poles being in contact with or impinging upon one another, which is generally undesirable, as discussed later.
Finally, the strict requirement for the positioning of the magnet poles also leads to a lack of flexibility in the design of machines according to U.S. Pat. No. 4,713,569. That is, the positions of the magnet poles cannot be altered in response to any other design consideration.